Twistor theory of symplectic manifolds

Abstract

This article is a contribution to the understanding of the geometry of the twistor space of a symplectic manifold. We consider the bundle $Z$ with fibre the Siegel domain Sp(2n,R)/U(n) existing over any given symplectic 2n-manifold M. Then, after recalling the construction of the almost complex structure induced on $Z$ by a symplectic connection on M, we study and find some specific properties of both. We show a few examples of twistor spaces, develop the interplay with the symplectomorphisms of M, find some results about a natural almost Hermitian structure on $Z$ and finally prove its n+1-holomorphic completeness. We end by proving a vanishing theorem about the Penrose transform.

34 pages

References in corpus (1)

• Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach

Cited by in corpus (4)

• On invariants of almost symplectic connections
• Hermitian and quaternionic Hermitian structures on tangent bundles
• On the twistor space of pseudo-spheres
• Remarks on symplectic twistor spaces